He is to lay his hand on the head of the burnt offering: The sacrificial animal represented the worshiper in the ceremony (Leviticus 4:4, 15, 24, 29, 33; 16:21). This act showed that the animal's death represented the owner's death. During the Day of Atonement ceremony, it showed that God transferred the owner's sins to the animal (16:20–22). In the peace offering (3:2, 8, 13), the act probably showed ownership of the sacrificed animal.
The phrase on his behalf may mean “as payment for sin” (sometimes called expiation) or “as a substitute for the sinner’s death” (sometimes called propitiation).
to make atonement for him: While older commentaries describe this as “covering over” sin, recent studies define it as “blotting out, erasing, or cleansing from” sin. The concept involves turning away God’s judgment against sin, making the worshiper clean in a ceremonial way, and restoring fellowship with God and with the community of God’s people.